Oddly, it seems Gmail is working, flud is connecting...

But web browsers and apps like Google Play can't connect.

Restarting the device hasn't worked.

Are there any ideas beyond a reset?

Edit:

This is SOLVED.

TL;DR: Anyone with a similar problem, check the system date and time.

Longer version:

It turns out that without a SIM card, and having run the battery down completely, once recharged and rebooted, the date and time were set back to a date in 2018, from before the tablet was even manufactured.

With the date/time so off, security certificate servers were rejecting my device, and a great many Android apps don't bother telling you the server's reason for rejection.
(You'd think Firefox, the Cadillac of browsers would, but it didn't.)

Anyway, thanks to everyone who upvoted, replied, or even just looked.